The spectral properties of \({\cal U}_{1/2}\)  and \({\cal U}_{1/3}\) 
are presented in Fig. REF . In Figs. REF (a)
and REF (c), we observe the emergence of states
at the fractional-quasienergies \(E=\pm \pi /2\)  and \(E=\pm \pi /3,\pm 2\pi /3\)  for the square-root and cubic-root systems respectively, whose
spatial profiles are found to be localized around the four corners
of the lattice. The numbers of these corner modes can further be controlled
by changing the real part of onsite potential \(u\) , as can be seen
clearly in Figs. REF (b) and REF (d).
The critical values \((u_{1},u_{2})\)  where the number of corner modes
change correspond to topological phase transition points, which are
determined by the gapless condition of the parent model [1]},
i.e.,
\(\cos \left[J_{1}\sqrt{1-(n\pi -u)^{2}/J_{2}^{2}}\right]\cosh v=\pm 1.\) 
